Theorem xrmaxleim 11207

Description: Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.)

Assertion

Ref	Expression
xrmaxleim	|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B -> sup ( { A , B } , RR* , < ) = B ) )

Proof of Theorem xrmaxleim

Dummy variables f g y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrlttri3 9754	. . . 4 |- ( ( f e. RR* /\ g e. RR* ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
2	1	adantl 275	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ ( f e. RR* /\ g e. RR* ) ) -> ( f = g <-> ( -. f < g /\ -. g < f ) ) )
3		simplr 525	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) -> B e. RR* )
4		prid2g 3688	. . . 4 |- ( B e. RR* -> B e. { A , B } )
5	3, 4	syl 14	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) -> B e. { A , B } )
6		simpllr 529	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> A <_ B )
7		xrlenlt 7984	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) )
8	7	ad3antrrr 489	. . . . . 6 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> ( A <_ B <-> -. B < A ) )
9	6, 8	mpbid 146	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> -. B < A )
10		breq2 3993	. . . . . . 7 |- ( y = A -> ( B < y <-> B < A ) )
11	10	notbid 662	. . . . . 6 |- ( y = A -> ( -. B < y <-> -. B < A ) )
12	11	adantl 275	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> ( -. B < y <-> -. B < A ) )
13	9, 12	mpbird 166	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = A ) -> -. B < y )
14		xrltnr 9736	. . . . . 6 |- ( B e. RR* -> -. B < B )
15	14	ad4antlr 492	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = B ) -> -. B < B )
16		breq2 3993	. . . . . . 7 |- ( y = B -> ( B < y <-> B < B ) )
17	16	notbid 662	. . . . . 6 |- ( y = B -> ( -. B < y <-> -. B < B ) )
18	17	adantl 275	. . . . 5 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = B ) -> ( -. B < y <-> -. B < B ) )
19	15, 18	mpbird 166	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) /\ y = B ) -> -. B < y )
20		elpri 3606	. . . . 5 |- ( y e. { A , B } -> ( y = A \/ y = B ) )
21	20	adantl 275	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) -> ( y = A \/ y = B ) )
22	13, 19, 21	mpjaodan 793	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) /\ y e. { A , B } ) -> -. B < y )
23	2, 3, 5, 22	supmaxti 6981	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ A <_ B ) -> sup ( { A , B } , RR* , < ) = B )
24	23	ex 114	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B -> sup ( { A , B } , RR* , < ) = B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703   = wceq 1348   e. wcel 2141   {cpr 3584   class class class wbr 3989   supcsup 6959   RR*cxr 7953   < clt 7954   <_ cle 7955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltirr 7886  ax-pre-apti 7889

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-iota 5160  df-riota 5809  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960

This theorem is referenced by:  xrmaxltsup  11221  xrmaxadd  11224  xrmineqinf  11232